Physicists Use Intuition Can We Formalize . . . An Example of . . . Why Are Infinities . . . Logarithms Are Not Infinity: This Infinity Problem . . . A Rational Physics-Related In Reality, Infinities . . . What Should We Do Explanation of the Since There Are No . . . But With Logarithms . . . Mysterious Statement Home Page by Lev Landau Title Page ◭◭ ◮◮ Francisco Zapata, Olga Kosheleva, and Vladik Kreinovich ◭ ◮ University of Texas at El Paso, El Paso, Texas 79968, USA, Page 1 of 14 fazg74@gmail.com, olgak@utep.edu, vladik@utep.edu Go Back Full Screen Close Quit
Physicists Use Intuition Can We Formalize . . . 1. Physicists Use Intuition An Example of . . . • Physicists have been very successful in predicting phys- Why Are Infinities . . . ical phenomena. This Infinity Problem . . . In Reality, Infinities . . . • Many fundamental physical phenomena can be pre- What Should We Do dicted with very high accuracy. Since There Are No . . . • The question is: how do physicists come up with the But With Logarithms . . . corresponding models? Home Page • In this, physicists often use their intuition. Title Page • This intuition is, however, difficult to learn, because it ◭◭ ◮◮ is not formulated in precise terms. ◭ ◮ • It is imprecise, it is intuition, after all. Page 2 of 14 Go Back Full Screen Close Quit
Physicists Use Intuition Can We Formalize . . . 2. Can We Formalize Physicists’ Intuition – at An Example of . . . Least Some of It? Why Are Infinities . . . • It would be great to be able to emulate at least some This Infinity Problem . . . of this intuition in a computer-based systems. In Reality, Infinities . . . What Should We Do • Then, the same successful line of reasoning can be used Since There Are No . . . to solve many other problems. But With Logarithms . . . • Computers, however, only understand precise terms. Home Page • So, to be able to emulate physicists’ intuition on a com- Title Page puter, we need describe it in precise terms. ◭◭ ◮◮ ◭ ◮ Page 3 of 14 Go Back Full Screen Close Quit
Physicists Use Intuition Can We Formalize . . . 3. An Example of Physicists’ Intuition: Landau’s An Example of . . . Statement about Logarithms Why Are Infinities . . . • Nobel-prize physicist Lev Landau often said that “log- This Infinity Problem . . . arithms are not infinity”. In Reality, Infinities . . . What Should We Do • This means that, in some sense, the logarithm of an Since There Are No . . . infinite value is not really infinite. But With Logarithms . . . • From the purely mathematical viewpoint, this state- Home Page ment by Landau makes no sense. Title Page • Of course, the limit of ln( x ) when x tends to infinity is ◭◭ ◮◮ infinite. ◭ ◮ • This was a statement actively used by a Nobel-prize Page 4 of 14 winning physicist. Go Back • So, we cannot just ignore it as a mathematically igno- rant nonsense. Full Screen Close Quit
Physicists Use Intuition Can We Formalize . . . 4. Why Are Infinities Important in the First Place? An Example of . . . • In physics, everything is finite, infinities are mathemat- Why Are Infinities . . . ical abstractions, what is the big deal? This Infinity Problem . . . In Reality, Infinities . . . • Let us compute the overall mass m of an electron. What Should We Do • According to special relativity theory, m = E/c 2 , where Since There Are No . . . E = m 0 · c 2 + E el . But With Logarithms . . . • Here E el is the energy of the electron’s electric field. Home Page • According to the same relativity theory, the speed of Title Page all communications is limited by the speed of light. ◭◭ ◮◮ • As a result, any elementary particle must be point- ◭ ◮ wise. Page 5 of 14 • Otherwise, different parts – due to speed-of-light bound Go Back – would constitute different sub-particles. Full Screen E ( x ) = c 1 · q • The electric field � E is � r 2 . Close Quit
Physicists Use Intuition Can We Formalize . . . 5. Why Are Infinities Important (cont-d) An Example of . . . • The field’s energy density ρ ( x ) is proportional to the Why Are Infinities . . . square of the field: ρ ( x ) = c 2 · ( � E ( x )) 2 . This Infinity Problem . . . In Reality, Infinities . . . • So, ρ ( x ) = c 3 · 1 def = c 2 · ( c 1 · q ) 2 . r 4 , where c 3 What Should We Do Since There Are No . . . • Thus, the overall energy of the electric field can be found if we integrate this density over the whole space: But With Logarithms . . . Home Page � ∞ 4 π · r 2 1 � � E el = ρ ( x ) dx = c 3 · r 4 dx = c 3 · dr = Title Page r 4 0 ◭◭ ◮◮ ∞ r 2 dr = − c 4 · 1 1 � � � c 4 · . � ◭ ◮ r � 0 Page 6 of 14 • For r = 0, we get a physically meaningless infinity! Go Back Full Screen Close Quit
Physicists Use Intuition Can We Formalize . . . 6. This Infinity Problem is Ubiquitous An Example of . . . • The problem is not just in the specific formulas for the Why Are Infinities . . . Coulomb law, the problem is much deeper. This Infinity Problem . . . In Reality, Infinities . . . • Many interactions are scale-invariant in the sense that What Should We Do they have no physically preferable unit of length. Since There Are No . . . • If we change the unit of length to a new one which is But With Logarithms . . . λ times smaller, then we get r ′ = λ · r . Home Page • Scale-invariance means that all the physical equations Title Page remain the same after this change. ◭◭ ◮◮ • Of course, we need to appropriately change the unit for measuring energy density, to ρ → ρ ′ = c ( λ ) · ρ . ◭ ◮ Page 7 of 14 • Suppose that in the original units, we have ρ ( r ) = f ( r ) for some function f . Go Back • Then in the new units, we will have ρ ′ ( r ′ ) = f ( r ′ ) for Full Screen the exact same function f ( r ). Close Quit
Physicists Use Intuition Can We Formalize . . . 7. Infinity Problem is Ubiquitous (cont-d) An Example of . . . • Here, ρ ′ = c ( λ ) · ρ and r ′ = λ · r , so c ( λ ) · ρ ( r ) = f ( λ · r ) Why Are Infinities . . . and c ( λ ) · f ( r ) = f ( λ · r ) . This Infinity Problem . . . In Reality, Infinities . . . • It is known that every measurable solution of this equa- tion has the form f ( r ) = c · r α for some c and α . What Should We Do Since There Are No . . . • Thus, ρ ( r ) = c · r α and therefore, the overall energy of But With Logarithms . . . the corresponding field is equal to Home Page � ∞ � ∞ � � c · r α dx = c · r α · 4 π · r 2 dr = c ′ · r 2+ α dr. ρ ( x ) dx = Title Page 0 0 ◭◭ ◮◮ • When α � = − 3, this integral is proportional to r 3+ α | ∞ 0 . ◭ ◮ • When α < − 3, this value is infinite at r = 0. Page 8 of 14 • When α > − 3, this value is infinite for r = ∞ . Go Back • In both cases, we get infinite energy. Full Screen • When α = − 3, the integral is proportional to ln( x ) | ∞ 0 . Close Quit
Physicists Use Intuition Can We Formalize . . . 8. Infinity Problem is Ubiquitous (cont-d) An Example of . . . • Logarithm is infinite for r = 0 (when it is −∞ ) and for Why Are Infinities . . . r = ∞ (when it is + ∞ ), so the difference is ∞ . This Infinity Problem . . . In Reality, Infinities . . . • The situation is not limited to our 3-dimensional proper What Should We Do space (corresponding to 4-dimensional space-time). Since There Are No . . . • It can be observed in space-time of any dimension d , But With Logarithms . . . where the area of the sphere is ∼ r d − 1 . Home Page • Thus the overall energy is proportional to the integral Title Page of r α · r d − 1 = r α + d − 1 . ◭◭ ◮◮ • If α � = − d , this integral is ∼ r α + d and infinite for r = 0 ◭ ◮ (when α < − d ) or for r = ∞ (when α > − d ). Page 9 of 14 • If α = − d , then the integral is proportional to ln( x ) | ∞ 0 Go Back and is, thus, infinite as well. Full Screen Close Quit
Physicists Use Intuition Can We Formalize . . . 9. In Reality, Infinities Are an Idealization An Example of . . . • In the above computations, we assumed that the dis- Why Are Infinities . . . tance r can take any value from 0 to infinity. This Infinity Problem . . . In Reality, Infinities . . . • In reality, the distance r cannot be too large: it cannot What Should We Do exceed the current radius R of the Universe. Since There Are No . . . • Similarly, the distance r cannot be too small. But With Logarithms . . . • When r 0 ≈ 10 − 33 cm, quantum effects become so large Home Page that the notion of exact distance becomes impossible. Title Page • When physicists talk about infinite values, they mean ◭◭ ◮◮ that the value is very large. ◭ ◮ • When physicists talk about 0 values, they mean that Page 10 of 14 the corresponding values are very small. • The quantum-effects distance 10 − 33 cm is much smaller Go Back than anything we measure. Full Screen • So, we can safely take this distance to be 0. Close Quit
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